Chapter 7, Problem 9RE

Solution

To calculate: The multiplication of the matrix and explain that the matrices are inverse.

They are inverses.

Given Information:

The matrix are defined as,

  $A=\left[\begin{array}{cccc}1& -2& 1& 1\\ 1& -1& 0& 3\\ 1& -1& 2& 2\\ 2& -4& 2& 3\end{array}\right],B=\left[\begin{array}{cccc}8& 1.5& 0.5& -4.5\\ 2& 0.5& 0.5& -1.5\\ -1& -0.5& 0.5& 0.5\\ -2& 0& 0& 1\end{array}\right]$

Formula Used:

Two matrices are said to be inverse if $AB=BA=I_n$ .

Where, the matrix $I_n$ is the identity matrix.

Calculation:

Consider the given matrices,

  $A=\left[\begin{array}{cccc}1& -2& 1& 1\\ 1& -1& 0& 3\\ 1& -1& 2& 2\\ 2& -4& 2& 3\end{array}\right],B=\left[\begin{array}{cccc}8& 1.5& 0.5& -4.5\\ 2& 0.5& 0.5& -1.5\\ -1& -0.5& 0.5& 0.5\\ -2& 0& 0& 1\end{array}\right]$

Find the multiplication.

  $\begin{array}{c}AB=\left[\begin{array}{cccc}1& -2& 1& 1\\ 1& -1& 0& 3\\ 1& -1& 2& 2\\ 2& -4& 2& 3\end{array}\right]\cdot \left[\begin{array}{cccc}8& 1.5& 0.5& -4.5\\ 2& 0.5& 0.5& -1.5\\ -1& -0.5& 0.5& 0.5\\ -2& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\text{From calculator}\right]\end{array}$

And,

  $\begin{array}{c}BA=\left[\begin{array}{cccc}8& 1.5& 0.5& -4.5\\ 2& 0.5& 0.5& -1.5\\ -1& -0.5& 0.5& 0.5\\ -2& 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{cccc}1& -2& 1& 1\\ 1& -1& 0& 3\\ 1& -1& 2& 2\\ 2& -4& 2& 3\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\text{From calculator}\right]\end{array}$

As the obtained matrix is the identity matrix of order 4 cross 4.

Hence, the given matrices are inverse.